Let $F,G:C\to D$ be naturally isomorphic functors. Taking the nerve, is $NF,NG:NC\to ND$ homotopy equivalent? Conversely, given a simplicial map $f:NC\to ND$, does there exists a functor $F:C\to D$ such that $NF=f$?
Yes to both. A natural transformation is a functor $C\times 2 \to D$, the nerve preserves products, and the nerve of $2$ is the 1-simplex. Geometric realisation gives homotopic maps. The second is elementary by the definition of nerve.